Laplace's Equation in 3-Dimension (Orthogonality and Eigen function)

Any help will be appreciated!
I am solving Laplace's Equation in 3-Dimensions using superposition.
The problem is that I don't know how to use orthogonality to solve for the constant when you have two Eigen functions.

This is my attempt to get a solution for v(x,y,z).

The solution using the given homogeneous boundary conditions comes out to be:

where and

The remaining boundary condition is:

Which gives:

How do I use orthogonality to find K when I have two eigen functions in the solution?

Just for your future reference, we can establish orthogonality between functions if we get <f,g> = 0 where <f,g> = Integral fg over the right interval.

Different inner products can have different intervals of integration and it depends on what your full space is (for example on space would be all functions on the interval [-1,1] while another might be all of R and subsequently every space has its own issues with how to construct a basis).

So with this if you have an inner product defined in terms of an L^2 integral inner product, you can check whether they are orthogonal and this is the kind of thing that happens in Fourier Analysis.